May  2012, 6(2): 121-130. doi: 10.3934/amc.2012.6.121

On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$

1. 

Department of Mathematics, KTH, S-100 44 Stockholm

2. 

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, I-06123 Perugia

Received  January 2011 Revised  August 2011 Published  April 2012

It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank $r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length $n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.
Citation: Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$. Advances in Mathematics of Communications, 2012, 6 (2) : 121-130. doi: 10.3934/amc.2012.6.121
References:
[1]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, The classification of some perfect codes, Des. Codes Cryptogr., 31 (2004), 313-318. doi: 10.1023/B:DESI.0000015891.01562.c1.  Google Scholar

[2]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, On the structure of symmetry groups of Vasilev codes, Probl. Inform. Transm., 41 (2005), 105-112. doi: 10.1007/s11122-005-0015-5.  Google Scholar

[3]

O. Heden, On the kernel of binary perfect 1-error correcting codes of length 15, manuscript, 33 pp., 1987. Google Scholar

[4]

O. Heden, A survey of perfect codes, Adv. Math. Commun., 2 (2008), 223-247. doi: 10.3934/amc.2008.2.223.  Google Scholar

[5]

O. Heden, F. Pasticci and T. Westerbäck, On the existence of extended perfect binary codes with trivial symmetry group, Adv. Math. Commun., 3 (2009), 295-309. doi: 10.3934/amc.2009.3.295.  Google Scholar

[6]

K. T. Phelps, A general product construction for error correcting Codes, SIAM J. Algebra Discrete Methods, 5 (1984), 224-228. doi: 10.1137/0605023.  Google Scholar

[7]

K. T. Phelps, O. Pottonen and P. R. J. Östergård, The perfect binary one-error-correcting codes of length 15: Part II properties, IEEE Trans. Inform. Theory, 56 (2010), 2571-2582. doi: 10.1109/TIT.2010.2046197.  Google Scholar

[8]

F. I. Solov'eva, "On Perfect Codes and Related Topics,'' Pohang, 2004. Google Scholar

[9]

V. A. Zinoviev, On generalized concatenated codes, in "Colloquia Math. Societ. Janos Bolyai, 16, Topics in Inform. Theory,'' Keszthely, Hungary, (1975), 587-592.  Google Scholar

[10]

V. A. Zinoviev, Generalized cascade codes, Probl. Inform. Transm., 12 (1976), 5-15.  Google Scholar

[11]

V. A. Zinoviev and D. A. Zinoviev, Binary perfect and extended perfect codes of length 15 and 16 with ranks 13 and 14 (in Russian), Problemy Peredachi Informatsii, 46 (2010), 20-24.  Google Scholar

show all references

References:
[1]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, The classification of some perfect codes, Des. Codes Cryptogr., 31 (2004), 313-318. doi: 10.1023/B:DESI.0000015891.01562.c1.  Google Scholar

[2]

S. V. Avgustinovich, O. Heden and F. I. Solov'eva, On the structure of symmetry groups of Vasilev codes, Probl. Inform. Transm., 41 (2005), 105-112. doi: 10.1007/s11122-005-0015-5.  Google Scholar

[3]

O. Heden, On the kernel of binary perfect 1-error correcting codes of length 15, manuscript, 33 pp., 1987. Google Scholar

[4]

O. Heden, A survey of perfect codes, Adv. Math. Commun., 2 (2008), 223-247. doi: 10.3934/amc.2008.2.223.  Google Scholar

[5]

O. Heden, F. Pasticci and T. Westerbäck, On the existence of extended perfect binary codes with trivial symmetry group, Adv. Math. Commun., 3 (2009), 295-309. doi: 10.3934/amc.2009.3.295.  Google Scholar

[6]

K. T. Phelps, A general product construction for error correcting Codes, SIAM J. Algebra Discrete Methods, 5 (1984), 224-228. doi: 10.1137/0605023.  Google Scholar

[7]

K. T. Phelps, O. Pottonen and P. R. J. Östergård, The perfect binary one-error-correcting codes of length 15: Part II properties, IEEE Trans. Inform. Theory, 56 (2010), 2571-2582. doi: 10.1109/TIT.2010.2046197.  Google Scholar

[8]

F. I. Solov'eva, "On Perfect Codes and Related Topics,'' Pohang, 2004. Google Scholar

[9]

V. A. Zinoviev, On generalized concatenated codes, in "Colloquia Math. Societ. Janos Bolyai, 16, Topics in Inform. Theory,'' Keszthely, Hungary, (1975), 587-592.  Google Scholar

[10]

V. A. Zinoviev, Generalized cascade codes, Probl. Inform. Transm., 12 (1976), 5-15.  Google Scholar

[11]

V. A. Zinoviev and D. A. Zinoviev, Binary perfect and extended perfect codes of length 15 and 16 with ranks 13 and 14 (in Russian), Problemy Peredachi Informatsii, 46 (2010), 20-24.  Google Scholar

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